Experimental verification of the metal flux enhancement in a mixture of two metal complexes: the Cd/NTA/glycine and Cd/NTA/citric acid systems

Experimental verification of the metal flux enhancement in a mixture

of two metal complexes: the Cd/NTA/glycine and Cd/NTA/citric acid

systems

J. P. Pinheiro,

J. Salvador,

E. Companys,

J. Galceran

and J. Puy*

Received 29th July 2009, Accepted 10th November 2009

First published as an Advance Article on the web 17th December 2009 DOI: 10.1039/b915486h

Rigorous computation of the metal flux crossing a limiting surface of a system that contains a mixture of 1 : 1 metal complexes under steady-state planar diffusion in a finite domain and under excess of ligand conditions predicts, for some cases, an enhancement of the metal flux with respect to that expected in a system with independent complexes. Indeed, the coupling of the dissociation kinetics of both complexes can yield higher metal fluxes than expected with important environmental implications. By using the voltammetric techniques AGNES and stripping chronopotentiometry, this paper provides experimental evidence of this enhancement for two systems: Cd/NTA/glycine and Cd/NTA/citric acid. The flux measured in both cases is in good agreement with the flux computed for the global system, exhibiting maximum enhancement ratios above 20%. Theoretical discussion of the flux enhancement factors and of the conditions for this enhancement are also provided.

1.

Introduction

The availability of metals to microorganisms or analytical sensors in natural systems is determined by a dynamic process that includes different steps, among which we find the inter-nalization of the metal at the sensor/microorganism surface and the transport and interaction with ligands, particles and colloids present in the media.1

Lability criteria predict which step is limiting the metal flux:2–8 either the dissociation, in which case the system is called partially labile or inert, or the transport to the surface, in which case the system is called labile. Moreover, the lability degree has also been introduced to indicate the percentage of the complex contribution to the uptake flux with respect to its maximum contribution obtained when the kinetics of the complexation processes were fast enough to reach equilibrium at any time and relevant spatial position.3,4,7,9–11 It has been shown that the lability degree depends on the kinetic constants, diffusion coefficients, size of the sensor, composition of the system, etc.

In a system with only one ligand, an increase of the ligand concentration decreases the lability degree of the complex due to the favouring of its association.3,12 Despite mixtures of ligands corresponding to the most common situation in natural media, mixture effects have only recently been described.13–17_{By rigorous simulation, it has been shown that}

the increase of the concentration of one ligand decreases the lability degree of its complex in the mixture as in the case of a single ligand system, but also influences the lability degree of the other complexes. Thus, interactions of the complexes in a mixture can play unexpected and relevant roles in the metal

availability. For instance, it has been claimed that the addition of a small amount of a labile complex to a system with an almost inert one gives rise to an enhancement of the metal flux. Different systems fulfilling these conditions have been analyzed by simulation and the enhancement has been justified by means of the reaction layer approximation which has been extended to mixtures of ligands.15–17 _{However, no experimental evidence}

has been reported until now of this flux enhancement. Few analytical techniques for trace metal speciation analysis allow the determination of dynamic parameters. Of these, in recent years, we highlight the remarkable development of electroanalytical methods such as stripping chronopotentio-metry, SCP,18 _{and scanned stripping chronopotentiometry,}

SSCP.19 An additional and independent measurement of the free metal concentrations can be obtained from the voltammetric stripping technique AGNES.20

It is the aim of this paper to provide experimental evidence of this enhancement by measuring the metal flux in different systems with a mercury electrode by using SSCP. Two systems have been analyzed: Cd/NTA/glycine and Cd/NTA/citric acid, as two examples of a fixed inert complex (CdNTA) with two different labile complexes. Together with the discussion of the experimental results in section 4, the paper provides in section 2 an approximate analytical expression for the metal flux and for the enhancement factor, while section 3 gathers experimental information.

2.

Theoretical background

2.1 The system and its rigorous solution

For simplicity, let us restrict ourselves to considering a solution with a mixture of only 2 independent ligands 1_L

and2L which can bind a metal ion M according to the scheme Mþi_{L ! } i_k a i_k d Mi_{L i¼ 1; 2 ð1Þ}

a_{IBB/CBME, Department of Chemistry, Biochemistry and Pharmacy,}

Faculty of Sciences and Technology, University of Algarve, 8005-139 Faro, Portugal

b

Departament de Quı´mica. Universitat de Lleida, Rovira Roure 191, 25198 Lleida, Spain. E-mail: [email protected]

where i_K, i_k

a and ikd are, respectively, the equilibrium and

the association and dissociation kinetic constants of the complexation process to the ligand iL. Let ci stand for the

concentration of species i, and let us assume that each ligand is present in the system in a great excess with respect to the metal, so that ci_L ci_Lis constant at any spatial point. The

corresponding equilibrium conditions read: i_K0_i_Kc i_L¼ i_k aci_L i_k d ¼ i_k0 a i_k d ¼c  Mi_L c M ð2Þ In the deposition stage of SSCP, stirring actually limits the region of interest for reaction–diffusion to just the diffusion layer which can be approximately taken as a constant and is denoted as g in this work. Considering diffusion towards a planar surface in a finite diffusion domain of thickness g, the rigorous metal flux for steady state conditions can be written as:4 JM¼ DM c M g þ X2 i¼1 D_Mi_L c Mi_L g i_{x ð3Þ}

where Di stands for the diffusion coefficient of species i.

Eqn (3) indicates that the total metal flux JMcan be

under-stood as the addition of different contributions of the metal species (free metal and the different complexes) present in the system. The parameter ix is called the lability degree of the complex i and takes values in the range 0oixo 1, so that D_Mi_L

c_{Mi L}

g is the maximum contribution of the complex M

i_L.

This maximum value reflects the transport limitation and is reached when there is no kinetic limitation in the dissociation of this complex.

i

x values can be obtained by solving a system of 3 linear diffusion equations corresponding to M, M1_{L and M}2_{L as a}

particular case of the general methodology developed for any number of ligands present.4,9,21,22 These results will here be referred to as ‘‘rigorous’’ simulation. When the complex does not contribute at all to the metal flux (ix = 0) it is called inert and JM¼ Jfree¼ DM

c M

g.

2.2 The mixture effect in a system of two complexes

The analysis in ref. 4 of the behaviour of a system that contains one metal and a mixture of ligands in steady state conditions showed that, in general, the addition of an inert ligand decreases the lability degree of all the complexes present in the system, while the addition of a labile ligand tends to increase the lability degree of all of them.

Of practical relevance is the quantification of the mixture effect in the metal flux. In order to measure this effect, let us define Jn=1M as the flux resulting from the

assumption that the complexes keep the lability degree that they would have in simpler systems with one complex at a time: J_Mn¼1¼ DMc  M g þ X2 i¼1 D_Mi_L c Mi_L g i_xn¼1 _ð4Þ

wherei_xn=1_{is the lability degree of complex i in a single ligand}

system. Notice that eqn (4) is formally identical to eqn (3) with only the change ofix toixn=1. Jn=1M corresponds, then, to the

value expected in a mixture of independent complexes, i.e.

without interaction effects between the respective lability degrees. As previously reported,3 i_xn=1 _{can be rigorously}

computed as: i_xn¼1_¼ iz tanhiz i_zþi_ei_Ktanhi_z ð5Þ whereie = DMiL/DMand i_{z¼ g} ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i_{kdð1 þ}i_ei_K0_Þ i_eD M s ð6Þ The evaluation of eqn (3) and (4) is performed using common values of the parameters, in particular, a ligand concentration equal to that of the mixture.

The parameter iz in eqn (6) can be thought of as (approximately) a normalized dimensionless inverse reaction layer, given that for the usual case of interest (ieiK0_c1):

i_{z g} ffiffiffiffiffiffiffiffi i_k0 a DM s ¼_ig m ð7Þ

A good approximation to eqn (4) is (ieiK0c1, g cim)

i_xn¼1_ g

gþi_ei_K0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_D M=ik0a

q ð8Þ

A system with only two complexes can be understood as the addition of the more labile to the more inert or just in the reverse order. Their mutual influence on the lability degree is opposite, so that the global effect at the level of the resulting metal flux is—due to partial cancellation—less pronounced than expected from the changes in the particular lability degrees. However, there are situations in which there is a non-negligible net result. Under these conditions, important deviations of the metal flux from the value expected from the lability degrees of the single ligand system can arise. For instance, if a ligand forming a labile complex is added to a system that contains an almost inert complex, even a small proportion of the labile complex can lead to an enhancement of the metal flux over the value expected from independent complexes. As we will see, under these conditions, we have an increase of the lability of the inert complex (by the interaction with the labile one) which increases the metal flux, while the effect of the inert one on the labile complex is negligible on the metal flux due to: (i) the low concentration of the labile complex and (ii) to the fully labile character of this complex which prevents its shifting towards a lower lability degree.

Quantitatively, the metal flux enhancement factor can be defined as:16,17

s¼ JM Jn¼1

M

ð9Þ

2.3 Analytical expressions for the enhancement factor 2.3.1 Rigorous expression for the limiting case of one fully labile complex in the mixture. A simple analytical expression for the metal flux can be obtained from the rigorous solution given in the ESI of ref. 4 by considering the limiting case where

one complex, for instance M2_{L, is fully labile (}2_k

d- N) while

the other is not: JM¼ DMcM g 1þ1_e1_K0_þ2_e2_K0   1þ 1e1_K0 ð1þ2_e2_K0_Þtanh z_z ð10Þ with z¼ g ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1_{kdð1 þ}1_e1_K0_þ2_e2_K0_Þ DM1_{eð1 þ}2_e2_K0_Þ s ð11Þ The corresponding lability parameters in the mixture are

1_x¼ 1 tanh z z 1þ 1e1_K0 ð1þ2_e2_K0_Þtanh z_z ; 2x¼ 1 ð12Þ

Eqn (10) leads to a very simple expression for the enhancement factor defined in eqn (9):

s¼ 1þ 1_e1_K0_þ2_e2_K0   1þ 1e1_K0 ð1þ2_e2_K0_Þ tanh z z   1þ1_e1_K0 1ztanh1_z 1_zþ1_e1_Ktanh1_zþ2e2K0   ð13Þ 2.3.2 Conditions for the maximum enhancement. In order to have a rough estimate of the conditions for which the enhancement is produced, we observe that the plot of the enhancement factor (Fig. 1) as given by eqn (13) exhibits a maximum. The position of this maximum will essentially correspond to the minimum of the denominator in eqn (13). Taking1_{e =}2_{e = 1, tanh z = 1, tanh}1_{z = 1, 1{}2_K0_{1_K0_,

1

K0c2K’z and1z{1K0, the denominator can be written as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1_K0_D M g2 1_kd2_Kc2 L s ð1zþ2_{K c} 2LÞ ð14Þ

The condition we seek is: d dc2_L 1_z ffiffiffiffiffiffiffi c2_L p þ2_K ffiffiffiffiffiffiffi_c 2_L p   ¼ 0 ð15Þ

so that the extreme condition can be written as 2_Kc

2_L¼1z

g

1_m ð16Þ

Expression (16) gives the concentration of the labile ligand, here labelled as2_{L, to be added into a solution of the metal}

and the almost inert ligand, here labelled as1L, to obtain the maximum enhancement effect.

With respect to c1L(the concentration of the inert ligand),

expression (13) indicates that s increases monotonically with increasing c1L. Although s could be huge for very large c1L, the

absolute value of the metal flux could be negligible.

In order to look for a physical interpretation of the condition for the maximum, we re-write eqn (16) as

D_M2_L c M2_L g  DM c_M 1_m ð17Þ

We have multiplied both terms by the equal diffusion coefficients (1e = 1) in order to obtain on the l.h.s. the flux of the labile complex and on the r.h.s. the flux of the inert complex. In this way, the maximum condition would stem from a similar contribution to the flux of the fully labile and inert complexes.

In practice, expression (16) could be useful as a quick guideline of the conditions suitable for the enhancement effect to be noticeable.

3.

Experimental

All solutions were prepared in ultrapure water from MilliQ Simplicity (resistivity 4 18 MO cm). Cd(II) stock solutions were prepared from dilution of cadmium standard solutions (1000 mg L1Merck), and the NaNO3used to adjust the ionic

strength solution was prepared from the solid (Merck, suprapur). Stock solution MOPS (3-(N-morpholino)propanesulfonic acid) buffer was prepared from the solid (Sigma-Aldrich, SigmaUltra 499.5%). HNO3 (Merck, suprapur) and KOH

(solid from Fluka, p.a.) solutions were used to adjust the pH. Nitrilotriacetic acid (NTA), glycine and citric acid were prepared from the solid (Fluka, puriss p.a.).

3.2 Electrochemical experiments

SCP/SSCP and AGNES experiments were performed in the same day using an Eco Chemie Autolab PGSTAT12 potentiostat in conjunction with a Metrohm 663VA stand and a personal computer using the GPES 4.9 software (Eco Chemie). Electrodes included a static mercury drop electrode (radius 1.78  104 _{m for the SCP/SSCP experiments and 1.41 }

104 m for the AGNES experiments, Fluka mercury p.a.) (working electrode), a saturated calomel electrode with a 0.10 M NaNO3salt bridge (reference electrode), and a glassy

carbon counter electrode. Measurements were performed at 25.0 1C in a reaction vessel thermostated with a bath (Selecta, Unitronic 100). A glass combined electrode (Orion 9103) was attached to a Thermo Orion 720Aplus ion analyzer to control the pH.

3.2.1 SSCP/AGNES calibration. SCP/SSCP and AGNES experiments (which were performed in the same electro-chemical cell) require a calibration plot at the same ionic strength (0.10M) as the main measurement. For AGNES, a three point calibration was performed (usually at 1.6, 3.2 and

Fig. 1 Enhancement factor (s) computed with eqn (13) for the Cd + NTA + citric system with parameters in Table 1 and cNTA=

5.0 104mol m3of total metal concentration) after which the SSCP calibration was done at the highest concentration. The calibrations were performed at low pH (o4) in order to avoid losses to the container walls.

3.2.2 SSCP and AGNES combined experiments. When both SSCP and AGNES techniques were used in the same solution, after the calibrations, NTA (2 103mol m3) was added directly to the calibration solution, together with MOPS pH buffer. The pH was adjusted to 8.00 0.05 and SSCP and AGNES data were acquired.

3.2.3 SCP/SSCP titrations. The process was parallel to the previous case, but only the SSCP calibration was performed. Then, the labile ligand was added in the range 5 102_to

4 mol m3for citric acid and from 1 101to 10 mol m3for glycine and the SCP data were acquired for each addition.

3.2.4 SCP/SSCP parameters. Stripping chronopotentio-metry techniques have two steps: deposition and quantification. During the deposition step (accumulation) the metal ions are reduced at a constant potential (–0.75 V for Cd(II) vs. SCE) well above the standard reduction potential of the metal for a given time interval, the deposition time, td(60 s in this work).

The quantification of the metal ion accumulation is performed during the so-called stripping step when the metal ion is oxidized by application of a constant oxidizing current, Is,

of 1 109A in quiescent solution until the potential reached a value sufficiently beyond the transition plateau (–0.40 V for Cd(II)). The Isvalue corresponds to conditions that approach

complete depletion (the product Ist is constant, for decreasing

Is values). The experimental signal is called the limiting

transition time (t*) calculated from the integral of the dt/dE vs. Ecurve obtained from the raw data (potential vs. time18_).

In the case of SSCP, a series of measurements is made over a range of deposition potentials, Ed. The transition time values

are then represented against the deposition potential, t vs. Ed.

The resulting plots, called SSCP curves, are formally equiva-lent to a voltammogram. The limiting transition time (t*, equivalent to the SCP result) and the half-wave potential (E1/2) are the most useful experimental parameters extracted

from these curves.

3.2.5 AGNES parameters. A DPP experiment with the largest mercury drop (radius 2.03 104m) was performed during the calibration, so that, from its peak potential, we could compute the E-value corresponding to the desired preconcentration factor or gain Y for any of the steps of the AGNES experiment.20The potential program for the AGNES experiment consisted in applying three potential steps:23

(i) E1,aunder reduction diffusion limited conditions,

corres-ponding to Y1,a= 1 108for a time t1,a(with stirring). The

suitable t1,adepends on the desired gain Y (or Y1,b) applied:

from previous experiments, it is known that t1,a= 35 s for

Y1,b= 50;

(ii) E1,bcorresponding to the desired Y1,bfor a t1,b longer

always than 3t1,a, (with stirring) and waiting time tw= 50 s

(without stirring). The value of Y1,b was selected to yield a

current above the limit of detection;

(iii) E2corresponding to Y2= 1 108under re-oxidation

diffusion limited conditions for 50 s, with the response current being read at t2= 0.20 s.

To subtract other components of the measured current different than the faradaic one, the shifted blank (see ref. 24) was performed with Y1= 0.01 (a negligible Y compared to the

Y1,bof the main measurement).

4.

Results and discussion

4.1 Retrieving information from SSCP/SCP

4.1.1 Stability constant. For the depletion mode of scanned stripping chronopotentiometry (SSCP)19 a known rigorous equation is available for the full wave in the kinetic current regime.25With this expression, the characteristic para-meters of the SSCP wave (t* and Ed,1/2) can be used to obtain

information on the lability of metal complex systems.26 The thermodynamic complex stability constant, K, can be calculated from the shift in the half-wave deposition potential, DEd,1/2, (analogous to the DeFord–Hume expression)

irrespec-tive of the degree of lability of the system:26 lnð1 þ K0Þ ¼ ðnF=RTÞDEd;1=2 lnðtMþL=t



MÞ ð18Þ where t_MþLand t_Mdenote the t values for limiting deposition current conditions in the presence and in the absence of ligand, respectively. The values obtained for Cd–NTA, Cd–citrate and Cd–glycine are reported in Table 1. The corresponding bulk free Cd concentration values were confirmed with AGNES experiments.

4.1.2 Kinetic constants. The limiting transition time is proportional to the deposited charge. In a system of one complex ML, the limiting deposition flux can be expressed in terms of the lability degree of this complex [see eqn (4)] as:

IStML¼ I  dtd¼ nFA DMcM g ð1 þ eK 0_xÞt d ð19Þ where I

dis the limiting deposition current and A is the area of the electrode. Once the value of x is found from eqn (19), the value of k0

acan be computed from eqn (4) or its approximation, eqn (8) (ieiK0_{c1, g c}i

m). Alternatively, one can proceed by simply dividing eqn (19) by its only metal limiting expression:

t MþL t

M

¼ 1 þ eK0x ð20Þ

From SSCP data in a solution with cT,NTA= 2 103mol m3

and cT,Cd = 5  104 mol m3, via expression (20), we

Table 1 Parameters used in the theoretical calculation of the fluxes and bulk concentrations. pH = 8; g = 2.2 105_m;1

e =2e = 1 NTA Citrate Glycine Protonation constants log (KH,1/m3mol1) 6.73 2.7 6.6 log (KH,2/m3mol1) 0.5 1.35 0.63 log (KH,3/m3mol1) 1.1 0.09 — Complexation constants log (KCd–L/m 3 mol1) 6.80 0.71 1.50 log (ka (Cd–L)/m3mol1s1) 6.26 7.42 5.79

obtained x, from which the effective or conditional association constant (k0

a) for Cd–NTA was computed with eqn (8) (see Table 1). Other kinetic constants follow from Eigen expression and are also reported in Table 1.

Note that recently, van Leeuwen et al.25analysed quantita-tively the impact of ligand protonation on metal speciation dynamics, showing that the metal complexation process to the different protonated species of the ligand can be reduced to only one complexation process with an effective association constant which depends on the intrinsic association constants of the various protonated forms and on the pH. Thus, results reported in Table 1 for the kinetic constants have to be considered as effective or conditional values for the ionic strength and pH considered in this work.

4.2 Experimental fluxes in mixtures

4.2.1 Cd/NTA/citric acid system. A replicate SSCP experi-ment with cT,NTA = 2  103 mol m3 and cT,Cd = 5 

104mol m3has been conducted by adding citric acid to the system in the concentration range cT,cit= 5 102mol m3

up to cT,cit = 1.6 mol m3. As the citric concentration

increases the metal flux in the deposition step also increases up to a factor of 3 as shown in Fig. 2 (see markers: and +). It could be thought that this increase is due to the shift of the bulk equilibrium towards the formation of Cd–citrate, a labile complex. However, a simple speciation calculation predicts that, although the free citrate concentration reached is higher than the NTA3concentration by 4 orders of magnitude, the species Cd–NTA is still the dominant complex (i.e. most abundant) sinceNTAK0_ccit

K0_.

Fig. 2 also shows a quite good reproducibility of the two replicate experiments and the agreement of these results with the theoretical ones obtained for the mixture system, either by rigorous numerical solution (continuous line), by assuming the citrate as fully labile and using eqn (10) (dotted line) or by using the reaction layer approximation15,16 which is here implemented by using eqn (15) from ref. 16,

(see dotted-dashed line). However, the theoretical results without considering the interaction between both complexes, i.e.calculated with eqn (4) and shown in Fig. 2 by a dashed line, clearly diverge from the experimental ones underestimating the flux. The difference between this dashed line and the experimental measurements corresponds to the enhancement term, so the existence of the metal flux enhancement due to the mixture effect is clearly evidenced in the figure.

The availability of the rigorous simulation tools allows for a detailed analysis of the enhancement conditions in the present system. The mutual influence of both complexes in the respective lability degrees is depicted in Fig. 3. As can be seen, the Cd–citrate complex is almost fully labile and the mixture does not modify noticeably the lability degree of this complex. Additionally, the Cd–NTA complex appears as almost inert with a lability degree slightly increasing as the concentration of the citric acid in the system increases. The influence of the changes of concentrations and lability degrees on the metal flux can be analyzed in Fig. 4. The contribution of the Cd–citrate complex to the global flux (see full squares), given by JCdcit¼ DCdcitc  Cdcit g Cdcit_{x ð21Þ}

increases almost linearly as the total concentration of citrate increases. However, JCd–citdoes not show any difference from

the Cd–citrate contribution expected in a system where the interaction between both complexes was frozen

J_Cdcitn¼1 ¼ DCdcitc  Cdcit

g

Cdcit_xn¼1 _ð22Þ

This behaviour is consistent with this complex being fully labile, both in the single ligand system and in the mixture (see Fig. 3), so that its contribution is not influenced by the presence of the NTA in the system.

The contribution of the Cd–NTA complexes to the metal flux is the most important one for cT,cit o 1.7 mol  m3

(see triangles in Fig. 4), and the impact of this influence in the mixture (in comparison with the single ligand system) is just

Fig. 2 Ratio of fluxes after and before the addition of different amounts of citrate to a system initially containing cT,NTA= 2 

103mol m3and cT,Cd= 5 104mol m3. Markers () and (+)

stand for two replicate series and the continuous line stands for the rigorous solution with parameters in Table 1. Dotted line corresponds to the limiting case of full lability of Cd–citrate [eqn (10)] and dashed line stands for the case of non-interacting complexes [eqn (4)]. Dotted-dashed line corresponds to the Zhang and Buffle approximation.15,16

Fig. 3 Lability degrees of complexes Cd + NTA + citrate along the titration corresponding to Fig. 2 with parameters in Table 1. Markers: full triangle forCdNTAx; open triangle for the hypothetical case with no interaction CdNTAxn=1; full square for Cd–citx; open square for the hypothetical case with no interaction Cd–cit_xn=1_{. Since} Cd–cit_{x E} Cd–cit

the responsible for the difference between JMand J n=1 M . Thus,

the increase of the lability degree of the almost inert complex, due to the addition into the system of the labile citric acid, albeit low, is the responsible of the enhancement of the metal flux. The enhancement factor s as defined in eqn (13) reaches 1.2

The concentration profiles can help in understanding the physical basis of the enhancement effect. Fig. 5 shows the concentration profiles of the metal and the Cd–NTA complexes at 3 points of the titration corresponding to concen-trations of added citric acid of 1, 2 and 8  101mol m3, respectively. The profile of the metal–citrate complex converges with the normalized metal profile beyond its small

particular reaction layer and it is not depicted in the figure. Only the profile of the Cd–NTA complex is depicted, as this complex is the responsible of the metal enhancement flux.

This figure deserves some comments: (i) the metal concen-tration profile is depleted as citrate is added into the system. This decrease in the Cd-profile favours the dissociation of Cd–NTA complex in all the diffusion domain, increasing its contribution to the metal flux as shown in Fig. 4 and being the way by which the flux enhancement is produced (i.e. the metal concentration profile can be seen as the coupling mechanism). (ii) Numerical simulations in the literature show that, due to the addition of the labile ligand into the system, the reaction layer of the inert one increases.4,16 _{As beyond the reaction}

layer the complex is in equilibrium with the metal, the net dissociation takes place within the reaction layer, so the increase of the thickness of this layer (due to the addition of the labile ligand) justifies the flux enhancement effect. No increase of the reaction layer thickness of the Cd–NTA is seen in Fig. 5 along the addition of citric acid into the system. Thus, a noticeable mixture effect can arise even when the change in the reaction layer is hardly detectable as in the present case. Notice that, in the present case, the reaction layer of Cd–NTA cannot increase since it extents over all the diffusion domain in Fig. 5.

(iii) The metal flux is (for finite dissociation rates of the involved complexes) proportional to the gradient of the free metal concentration at the electrode surface, so that the gradient of the metal profile should be larger as the citrate concentration increases (see Fig. 5). As this is hardly seen in the main picture of Fig. 5, a magnification of the profiles close to the electrode surface up to distances of the order of the composite reaction layer of the Cd–citrate complex is included as inset of Fig. 5. This magnification allows us to clearly see a crossing of the metal concentration profiles, so that the lowest metal concentration profile in the main figure (corresponding to a citrate concentration of 0.8 mol m3) decreases slowly until it steeply falls to the zero value within a close approach of the electrode, so that it yields the highest metal gradient at the electrode surface.

The application of condition (16) with the parameters of the citrate titration predicts a maximum (see Fig. 1) at c2L= ccitE

1.124 mol m3 which is a rough approximation to the real position of the maximum c2L = ccit E 0.847 mol m3. The

corresponding s-values [computed with eqn (13)] are 1.313 (for the true maximum) and 1.279 [using the approximate c2Lfrom eqn (16)].

4.2.2 The Cd/NTA/glycine system. In two experiments that cover different glycine concentration ranges, glycine has been added into a system with cT,Cd= 5 104mol m3and

cT,NTA = 2  103mol m3 in the range cT,Gly = 0.1 . . .

9.3 mol m3. Increasing the glycine concentration, the metal flux increases (Fig. 6). The main result of this figure is that there is also a good convergence of the theoretical results of the mixture system (at the rigorous level, by using eqn (10) or by using the reaction layer approximation, eqn (15) in ref. 16) with the experimental ones, while the theoretical results given by eqn (4) that neglect the interaction between the complexes diverge yielding lower flux values. An enhancement of the

Fig. 4 Fluxes and their components along the titration corres-ponding to Fig. 2 with parameters in Table 1. Solid line stands for the rigorous solution, dotted line stands for fluxes without interaction and dashed line for the free metal contribution. Markers: full diamond for the rigorous total flux, JM; open diamond for

the hypothetical total flux when there is no interaction between complexes Jn=1M [eqn (4)]; asterisk for free metal component,

Jfree¼ DMcM=g; full triangle for the flux associated to the complex

CdNTA, JCdNTA¼ DCdNTAcCdNTACdNTAx=g; full square for the flux

JCdCit¼ DCdCitcCdCitCdCitx=g; open triangle for the hypothetical flux

with no interaction Jn¼1

CdNTA¼ DCdNTAcCdNTACdNTAxn¼1=g; and open

square (coinciding with the full squares in this figure) for the hypothetical flux with no interaction Jn¼1

CdCit¼ DCdcitcCdCitCdCitx n¼1_=g.

Fig. 5 Normalised concentration profiles for the free metal (thick lower lines) and for the CdNTA (thin upper lines). Fixed cT,NTA= 2

103mol m3and cT,Cd= 5 104mol m3. Dotted lines stand for

cT,cit= 0.1 mol m3; dashed lines for cT,cit = 0.2 mol m3; and

continuous lines cT,cit= 0.8 mol m3. The inset shows the crossing of

metal flux is then evidenced in the figure. As the conditional stability constantCdNTA_K0_{is higher than}CdGly_K0_{, Cd ions are} preferentially bound to NTA even at the highest glycine concentration, being the concentration of Cd bound to glycine rather negligible.

Fig. 7 depicts the change of the lability degrees. Notice that now a decrease of the lability of the more labile complex is seen in the figure due to the presence of NTA in the system, while the lability degree of Cd–NTA increases due to the addition of glycine into the system.

However, the difference between JCdGly and Jn=1CdGly is

negligible (see Fig. 8) and again, the enhancement of the metal flux is due to the enhancement of JCdNTAin the interacting

mixture.

The concentration profiles of the free Cd and the Cd–NTA complex are similar to those reported in Fig. 5 and are not included here.

With the parameters of the glycine titration (used in Fig. 6), the maximum enhancement factor appears at c2L= cGlyE

0.137 mol m3 (where s = 1.312) while the approximation (16) yields c2L= cGlyE 0.182 mol m3(with a s = 1.306

computed with eqn (13)), which means a good approximation for the s-value.

5.

Conclusion

The lability degree of a complex is not an intrinsic property of this complex, but a property influenced by the concentration of the rest of complexes of this metal that exist in the solution. As a consequence of this influence, an enhancement of the limiting metal flux crossing a consuming surface was predicted by numerical simulation when a labile ligand is added into a system with an almost inert complex.

This prediction is here experimentally measured for the Cd/NTA/glycine and Cd/NTA/citric acid systems. Cd–NTA complexes behave as almost inert complexes in SCP experiments under the concentrations reported in the present work. Cd–glycine and Cd–citric behave as labile complexes under the same conditions. The enhancement factor reaches 20% of the metal flux for the case of the Cd–NTA–citrate mixture.

A detailed analysis has shown that, for both cases, the enhancement is due to the increase of the lability of the Cd–NTA complexes when glycine or citric acid is added. As a general mechanism, this increase in the lability degree can be related to the depletion of the metal concentration profile when the labile ligand is added into the system.

Simple approximate analytical expressions for the enhance-ment factor have been reported. These expressions have been used to obtain an estimation of the concentration of the labile ligand to be added to obtain the highest enhancement factor. These estimations are in agreement with the experimental measurements.

Fig. 6 Ratio of fluxes after and before the addition of different amounts of glycine to a system initially containing cT,NTA= 2 

103mol m3and cT,Cd= 5 104mol m3. Markers () and (+)

stand for two replicate series and the continuous line stands for the rigorous solution with parameters in Table 1. Dotted line corresponds to the limiting case of full lability of CdGly [eqn (10)] and dashed line stands for the case of non-interacting complexes [eqn (4)]. Dotted-dashed line corresponds to the Zhang and Buffle approximation.15,16

Fig. 7 Lability degrees of complexes Cd + NTA + glycine along the titration corresponding to Fig. 6 with parameters in Table 1. Markers: full triangle forCdNTAx; open triangle for the hypothetical case with no interaction CdNTA_xn=1_{; full square for} CdGly_{x; open square for the}

hypothetical case with no interactionCdGly_xn=1_.

Fig. 8 Fluxes and their components along the titration corresponding to Fig. 6 with parameters in Table 1. Lines as in Fig. 4. Markers: full diamond for the rigorous total flux, JM; open diamond for the

hypothetical total flux when there is no interaction between complexes Jn=1

M [eqn (4)]; asterisk for free metal component, Jfree¼ DMcM=g;

full triangle for the flux associated to the complex CdNTA, JCdNTA¼ DCdNTAcCdNTACdNTAx=g; full square for the flux

JCdGly¼ DCdGlycCdGlyCdGlyx=g; open triangle for the hypothetical

flux with no interaction Jn¼1

CdNTA¼ DCdNTAcCdNTACdNTAx

n¼1_{=g; and}

open square for the hypothetical flux with no interaction J_CdGlyn¼1 ¼ DCdGlycCdGlyCdGlyx

Acknowledgements

J.P.P. acknowledges the funding received by ‘‘IBB/CBME, LA, FEDER/POCI 2010’’ and a sabbatical grant SFRH/ BSAB/855/2008 from Fundac¸a˜o para a Cieˆncia e Tecnologia, Portugal. This work was also financially supported by the Spanish Ministry of Education and Science (Projects CTQ2006-14385 and CTM2006-13583) and from the ‘‘Comissionat per a Universitats i Recerca del Departament d’Innovacio´, Universitats i Empresa de la Generalitat de Catalunya’’.

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